Introduction to Probability and Statistics for Engineers and Scientists

446 Chapter 10:Analysis of Variance

TABLE 10.1 Values of Fr,s,.05

r= Degrees of Freedom

s= Degrees of for the Numerator

Freedom for the

Denominator 1 2 3 4

4 7.71 6.94 6.59 6.39

5 6.61 5.79 5.41 5.19

10 4.96 4.10 3.71 3.48

To determine how largeTSneeds to be to justify rejectingH 0 , we use the fact that
it can be shown that ifH 0 is true thenSSbandSSW are independent. It follows from
this that, whenH 0 is true,TShas anF-distribution withm−1 numerator andnm−m
denominator degrees of freedom. LetFm−1,nm−m,αdenote the 100(1−α) percentile of
this distribution — that is,

P{Fm−1,nm−m>Fm−1,nm−m,α}=α

where we are using the notationFr,sto represent anF-random variable withrnumerator
andsdenominator degrees of freedom.
The significance levelαtest ofH 0 is as follows:

reject H 0 if

SSb/(m−1)

SSW/(nm−m)

>Fm−1,nm−m,α

do not reject H 0 otherwise

A table of values ofFr,s,.05for various values ofrandsis presented in Table A4 of the
Appendix. Part of this table is presented in Table 10.1. For instance, from Table 10.1 we
see that there is a 5 percent chance that anF-random variable having 3 numerator and 10
denominator degrees of freedom will exceed 3.71.
Another way of doing the computations for the hypothesis test that all the population
means are equal is by computing thep-value. If the value of the test statistic isTS=v,
then thep-value will be given by

p-value=P{Fm−1,nm−m≥v}

Program 10.3 will compute the value of the test statisticTSand the resultingp-value.

EXAMPLE 10.3a An auto rental firm is using 15 identical motors that are adjusted to run
at a fixed speed to test 3 different brands of gasoline. Each brand of gasoline is assigned to
exactly 5 of the motors. Each motor runs on 10 gallons of gasoline until it is out of fuel.